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In order to properly service the movement you absolutely should remove the wheel so that you can clean any contaminated or degraded oil away (which if left could very easily result in premature wear and failure even if the side shake is acceptable now) and relubricate with fresh oil.
A dedicated 5 spoke wheel puller is the best way to go as it pulls at the hub, eliminating the risk of deforming the wheel, but not the only way. The important thing is to pull the wheel straight up with no twisting or canting of the wheel since the arbor is brittle and will snap sooner than bend.
This can be achieved using two thin blades worked under the wheel from opposite sides to gradually wedge the wheel up.
You may need to protect the bridge with some paper or tape to prevent scratches, and you need to use blades that are only very slightly thicker than the gap between the wheel and the bridge because you need to progress slowly, and don't be tempted to twist the blades. As long as the force is straight up in line with the arbor, and gradual enough to not deform the wheel, you should be alright.

What is the brand of the crystal. That looks like a low dome. I normally use GS crystals and the left one looks like a dive tite or one of “tite” models. Was there a tension ring. I never really order from cousins because shipping takes forever to USA, so not sure what they sell.